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Mirrors > Home > MPE Home > Th. List > Mathboxes > riotasv2s | Structured version Visualization version GIF version |
Description: The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5400) in the form of a substitution instance. Special case of riota2f 7386. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) |
Ref | Expression |
---|---|
riotasv2s.2 | ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) |
Ref | Expression |
---|---|
riotasv2s | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpc 1150 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → (𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑))) | |
2 | simp1 1136 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐴 ∈ 𝑉) | |
3 | riotasv2s.2 | . . . . . 6 ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) | |
4 | nfra1 3281 | . . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) | |
5 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
6 | 4, 5 | nfriota 7374 | . . . . . 6 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) |
7 | 3, 6 | nfcxfr 2901 | . . . . 5 ⊢ Ⅎ𝑦𝐷 |
8 | 7 | nfel1 2919 | . . . 4 ⊢ Ⅎ𝑦 𝐷 ∈ 𝐴 |
9 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑦 𝐸 ∈ 𝐵 | |
10 | nfsbc1v 3796 | . . . . 5 ⊢ Ⅎ𝑦[𝐸 / 𝑦]𝜑 | |
11 | 9, 10 | nfan 1902 | . . . 4 ⊢ Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑) |
12 | 8, 11 | nfan 1902 | . . 3 ⊢ Ⅎ𝑦(𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) |
13 | nfcsb1v 3917 | . . . 4 ⊢ Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶 | |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶) |
15 | 10 | a1i 11 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦[𝐸 / 𝑦]𝜑) |
16 | 3 | a1i 11 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
17 | sbceq1a 3787 | . . . 4 ⊢ (𝑦 = 𝐸 → (𝜑 ↔ [𝐸 / 𝑦]𝜑)) | |
18 | 17 | adantl 482 | . . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → (𝜑 ↔ [𝐸 / 𝑦]𝜑)) |
19 | csbeq1a 3906 | . . . 4 ⊢ (𝑦 = 𝐸 → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶) | |
20 | 19 | adantl 482 | . . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶) |
21 | simpl 483 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 ∈ 𝐴) | |
22 | simprl 769 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐸 ∈ 𝐵) | |
23 | simprr 771 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → [𝐸 / 𝑦]𝜑) | |
24 | 12, 14, 15, 16, 18, 20, 21, 22, 23 | riotasv2d 37815 | . 2 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝐴 ∈ 𝑉) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶) |
25 | 1, 2, 24 | syl2anc 584 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ∀wral 3061 [wsbc 3776 ⦋csb 3892 ℩crio 7360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-riotaBAD 37811 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-riota 7361 df-undef 8254 |
This theorem is referenced by: (None) |
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